Modular Exponentiation Calculator

This calculator computes modular exponentiation, a base raised to an exponent and then reduced modulo a number, which is written as base to the power exp, mod m. This operation is the computational heart of modern cryptography, including the RSA and Diffie-Hellman algorithms that secure the internet, as well as a fundamental tool in number theory. The challenge is that the exponent can be enormous, and raising a number to a huge power would create a value with thousands of digits, far too large to handle directly. The clever solution is fast modular exponentiation, also called exponentiation by squaring, which keeps the numbers small by taking the modulus at every step and squaring its way up through the exponent's binary representation. This means even astronomically large exponents are handled almost instantly. This tool implements it. You enter the base, the exponent, and the modulus, and the calculator returns the result, the remainder when the base raised to the exponent is divided by the modulus, along with the inputs for reference. It uses big-integer arithmetic internally, so the result is exact even for large values. The results update as you type. Use it for cryptography study and exercises, for number theory, for checking modular arithmetic, or for understanding how public-key encryption performs its calculations. The method works by writing the exponent in binary and repeatedly squaring the base modulo m, multiplying the running result whenever a binary digit is one, so the number of operations grows only with the number of digits in the exponent, not its size. This efficiency is what makes public-key cryptography practical. The modulus must be a positive whole number, and the result is always a non-negative integer less than the modulus.

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Uses fast exponentiation by squaring with big-integer arithmetic, so very large exponents compute instantly. The result is a non-negative integer less than the modulus.